QUALITY CONTROL 729 



Specification Type IV: The expected or average values of quality and 

 standard deviation shall he X' and a' respectively. The expected modal and 

 average qualities shall coincide and the probability function for the con- 

 stant system of causes shall be monotonically decreasing for all values 

 of X where x is measured from the mean. 



In this case the lower bound to the probability Par' is given by the 

 expression ^ 



P„.>1-^,- ■ (3) 



The limits can be obtained just as in case of Type III except that we 

 use point B in Fig. 4 instead of point A. It may be easily verified by 

 this nomogram that the Type IV values of Li and L„ for the special 

 problem considered for Type III are Li = 10.4 ohms and L„ = 0.33 

 ohm respectively. 



This shows that the additional requirements placed upon Type IV 

 over those of Type III make for better control in the sense that the 

 associated sampling limits are thereby decreased. By going further in 

 adding restrictions upon the cause system, we gain even more marked 

 improvements in the condition for control. In fact it is the system now 

 to be described that has been found to be the most useful practical 

 standard where the quality is measured as a variable. 



Specification Type V: The system of causes shall yield a product 

 distributed according to the Gram Charlier series ^ with arithmetic mean 

 X' , standard deviation a' , skewness k' and kurtosis ^2'. 



With the use of the four parameters we can detect lack of control 

 of product through the failure of the observed value of any parameter 

 determined from a sample of size n to fall within its sampling limits. 

 It may happen that lack of control will be indicated by deviation be- 

 yond the sampling limits for only one of the four parameters. This 

 case has already been illustrated in the article referred to in footnote 1. 

 We shall now present, however, a method of setting these limits which 

 is very easily applied. 



As a specific example, let us assume the following expected values: 



X' = 0, 

 c' = 1, 

 k' = 0, 



•^ Camp, Burton H., "A New Generalization of Tchebycheff's Statistical In- 

 equality," Bulletin of the Amer. Math. Soc, December 1922, pp. 427-432. Eq. 3 is 

 a special case of the general theorem of Camp. This theorem may be extended to 

 determine lower bound to the probability of an error of the average as is done in 

 this paper. 



' Of course we might use certain other functions involving the same parameters. 



